Gravity Seven Gaps Simplicial Class Simplicial Complex Card Pos

The simplicial subclass

A simplicial complex is a way of assembling simple pieces, such as points, edges, and tetrahedra, so that they fit together cleanly. In the Recognition Science framework's library, a machine-checked collection of formal theorems, the declaration simplicialComplex_card_pos establishes a basic fact about a specific class of these objects: for any bound B, the number of simplicial complexes with at most B edges is strictly greater than zero.

The theorem is proved in the Lean proof assistant, meaning it is verified by a computer kernel and carries no unproven assumptions. The proof of positivity is direct: the class contains at least one object, so its cardinality cannot be zero. To ensure this is not a vacuous truth about an empty collection, the library also constructs a concrete example, a single tetrahedron with all six of its edges. This witness, named oneTetComplex, is used to show that the positivity claim rests on a real object, not just on the empty complex.

The simplicial class itself is carved out by four conditions. Edges must have distinct endpoints, no two edges may connect the same pair of vertices, each tetrahedron must have four distinct vertices, and every pair of vertices in a tetrahedron must be joined by an edge. These conditions define what it means for a bounded incidence configuration to be a true abstract simplicial 3-complex. The framework models this as a subtype of a larger, garbage-inclusive class called BoundedComplex, which also contains degenerate or malformed configurations.

The theorem does not claim that every bounded complex is simplicial, nor does it assert anything about the number of simplicial complexes beyond positivity. It does not establish that the class is large or that it contains any particular complex beyond the single tetrahedron witness. The proof also does not address triangle faces, because the underlying data structure only carries vertices, edges, and tetrahedra; triangle closure is not expressible in this setting. The library records this limitation explicitly in a status structure, noting that triangle closure is not part of the current model.

What the theorem changes is the foundation for later work: it guarantees that the simplicial subclass is a meaningful, nonempty object of study. Any future argument that sums over all simplicial complexes, such as the path-sum measure Zsimp, can rely on the fact that the indexing set is finite and nonempty. This makes the positivity theorem a small but necessary step for building a well-defined measure over the simplicial subclass.

/-- **THEOREM (positive count).**  The simplicial subclass is nonempty for
every size cap, so the restricted path sum has a nontrivial configuration
space. -/
theorem simplicialComplex_card_pos (B : ℕ) :
    0 < Fintype.card (SimplicialComplex B) :=
  Fintype.card_pos

/-- **THEOREM (positive count).**  The simplicial subclass is nonempty for
every size cap, so the restricted path sum has a nontrivial configuration
space. -/
theorem simplicialComplex_card_pos (B : ℕ) :
    0 < Fintype.card (SimplicialComplex B) :=
  Fintype.card_pos

/-- **THEOREM (non-vacuous simplicial witness).**  The single-tetrahedron
complex is simplicial: distinct edge endpoints, no multi-edges, injective
corners, and every corner pair realized by one of the six skeleton edges.
Kernel-checked by `decide` on the finite index types (`Fin 4`, `Fin 6`,
`Fin 1`); no `native_decide`. -/
theorem oneTetComplex_isSimplicial : IsSimplicial oneTetComplex := by
  decide

/-- **THEOREM (positive count).**  The simplicial subclass is nonempty for
every size cap, so the restricted path sum has a nontrivial configuration
space. -/
theorem simplicialComplex_card_pos (B : ℕ) :
    0 < Fintype.card (SimplicialComplex B) :=
  Fintype.card_pos

What this page does not claim

The theorem does not establish that the simplicial class contains any complex other than the single tetrahedron witness. The theorem does not assert anything about the number of simplicial complexes beyond the fact that it is positive. The proof does not address triangle faces, because triangle closure is not expressible in the current data structure.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/SevenGaps/SimplicialClass.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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